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PERIODIC MOTION                           329
mass.   Find the period with which the particle will vibrate when displaced along the string.
Let L' be the stretched length of the string, A the area of its cross-section, q the distance of the particle from its position of equilibrium, and T\ and jP2 the tensile forces of the two parts of the string. Then by Hooke's law we have
£'      \    L
1     y   I         n              T t         7"       !      O   ~
T = X -----r-A               L
2 __ , L' -L~-2q \
2 Therefore the resultant force on the particle is
T: A
=      -i^ -- ITg==~~L" where X' = ^LX.   Hence the potential energy equals
But since the kinetic energy is given by
1           2X'
we obtain                       E = - mq* + — q2
Al                                 JLJ
for the total energy of the system.   Differentiating the last equation we get                                     mg + ^-o
which gives                                       ___
q^asm^—tt + to)
and                                P =
3. A cylinder performs small oscillations inside of a fixed cylinder. Find the period of the motion, supposing the contact between the cylinders to be rough enough to prevent sliding.